[FOM] Questions about existence of big sets in Z and ZF

[Zuhair Abdul Ghafoor Al-Johar]

1. In Z-Reg. can we have a set K such that every member of K is an equivalence class of subsets of K under bijection, and every subset of K is a member of a member of K, and for every well founded set x there is a member of K that has a member of it that is equinumerous to x?

2.In ZF-Reg. can we have a set S such that for every well founded set x
there is a surjection from a subset of S to x?

Zuhair